Distributive Property: Expanding 6(9a + 2b)

by Alex Johnson 44 views

Let's dive into how to expand expressions using the distributive property. It's a fundamental concept in algebra, and mastering it will help you simplify and solve more complex equations. In this article, we'll break down the expression $6(9 a+2 b)$ step by step, ensuring you understand the process thoroughly. So, grab your pencil and paper, and let's get started!

Understanding the Distributive Property

The distributive property is a cornerstone of algebra that allows us to multiply a single term by two or more terms inside a set of parentheses. The basic idea is that you "distribute" the term outside the parentheses to each term inside. Mathematically, it's expressed as:

a(b+c)=ab+aca(b + c) = ab + ac

Where 'a' is the term outside the parentheses, and 'b' and 'c' are the terms inside. You multiply 'a' by 'b' and then 'a' by 'c', and add the results together. This property is incredibly useful for simplifying expressions and solving equations. Without it, many algebraic manipulations would be impossible. For instance, when dealing with more complex polynomials or rational expressions, the distributive property is often the first step in simplifying the expression to a more manageable form. Furthermore, understanding the distributive property is crucial for grasping more advanced algebraic concepts, such as factoring and expanding binomials. It acts as a building block, providing the foundation for more sophisticated problem-solving techniques. By mastering this property, you'll find that many algebraic challenges become much easier to tackle. It is also used extensively in calculus and other higher-level mathematics, making it an indispensable tool in your mathematical toolkit. So, let's solidify our understanding and move on to applying it to a specific example.

Applying the Distributive Property to $6(9 a+2 b)$

Now, let's apply the distributive property to the expression $6(9 a+2 b)$. Here, 6 is the term outside the parentheses, and $9a$ and $2b$ are the terms inside. According to the distributive property, we need to multiply 6 by each term inside the parentheses:

6(9a+2b)=6∗(9a)+6∗(2b)6(9 a+2 b) = 6 * (9a) + 6 * (2b)

Let's break this down further. First, we multiply 6 by $9a$:

6∗(9a)=54a6 * (9a) = 54a

Then, we multiply 6 by $2b$:

6∗(2b)=12b6 * (2b) = 12b

Now, we add these two results together:

54a+12b54a + 12b

So, the expanded form of $6(9 a+2 b)$ is $54 a+12 b$. This means that option A, $54 a+12 b$, is the correct answer. The distributive property allows us to break down complex expressions into simpler terms, making them easier to understand and work with. In this case, we transformed a single term multiplied by a binomial into a sum of two terms. This process is fundamental in algebra and is used extensively in various mathematical contexts, from solving equations to simplifying expressions. Remember, the key is to systematically multiply the term outside the parentheses by each term inside, paying close attention to the signs and coefficients. With practice, you'll become more comfortable and efficient in applying the distributive property, which will greatly enhance your problem-solving skills in algebra and beyond. Now, let's take a look at some common mistakes to avoid when using the distributive property.

Common Mistakes to Avoid

When using the distributive property, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  1. Forgetting to Distribute to All Terms: Make sure you multiply the term outside the parentheses by every term inside. A common mistake is to only multiply by the first term and forget the rest. For example, in the expression $a(b + c)$, you must multiply 'a' by both 'b' and 'c'.
  2. Sign Errors: Pay close attention to the signs of the terms. If you're distributing a negative number, remember that multiplying by a negative changes the sign of the term. For example, $-a(b - c) = -ab + ac$. Notice how the $-c$ becomes $+ac$ when multiplied by $-a$.
  3. Incorrect Multiplication: Double-check your multiplication. Simple arithmetic errors can lead to incorrect results. Always ensure that you are multiplying the coefficients correctly. For example, $3(2x + 4) = 6x + 12$. Make sure $3 * 2 = 6$ and $3 * 4 = 12$.
  4. Combining Unlike Terms: After distributing, you can only combine like terms. Like terms have the same variable raised to the same power. For example, you can combine $3x$ and $5x$ to get $8x$, but you cannot combine $3x$ and $5x^2$.
  5. Not Simplifying Completely: After distributing and combining like terms, make sure your expression is fully simplified. This means there should be no more like terms to combine. For example, if you end up with $2x + 3x + 5$, you should simplify it to $5x + 5$.

By being mindful of these common mistakes, you can avoid errors and ensure that you're applying the distributive property correctly. Practice and attention to detail are key to mastering this important algebraic concept. Let's move on to some additional practice problems to further solidify your understanding.

Additional Practice Problems

To solidify your understanding of the distributive property, let's work through a few more practice problems:

  1. Problem: Expand $3(4x - 2y)$

    Solution: $3(4x - 2y) = 3 * (4x) - 3 * (2y) = 12x - 6y$

  2. Problem: Expand $-2(5a + 3b - c)$

    Solution: $-2(5a + 3b - c) = -2 * (5a) - 2 * (3b) + 2 * (c) = -10a - 6b + 2c$

  3. Problem: Expand $x(2x + y)$

    Solution: $x(2x + y) = x * (2x) + x * (y) = 2x^2 + xy$

  4. Problem: Expand $4p(3p - 2q + 5)$

    Solution: $4p(3p - 2q + 5) = 4p * (3p) - 4p * (2q) + 4p * (5) = 12p^2 - 8pq + 20p$

  5. Problem: Expand $-a(-b + 2c - 3d)$

    Solution: $-a(-b + 2c - 3d) = a * (b) - a * (2c) + a * (3d) = ab - 2ac + 3ad$

Working through these examples will give you more confidence in applying the distributive property. Remember to take your time, pay attention to the signs, and double-check your work. The more you practice, the more natural this process will become. With each problem, you're reinforcing your understanding and improving your skills. Keep practicing, and you'll master the distributive property in no time!

Conclusion

In conclusion, the distributive property is a fundamental concept in algebra that allows us to simplify expressions by multiplying a term by multiple terms inside parentheses. We've seen how to apply it to the expression $6(9 a+2 b)$, arriving at the expanded form $54 a+12 b$. We've also discussed common mistakes to avoid and worked through additional practice problems to solidify your understanding. By mastering the distributive property, you'll be well-equipped to tackle more complex algebraic challenges. Keep practicing, and you'll find that algebra becomes much more manageable and even enjoyable! For further learning, you can visit Khan Academy's article on the Distributive Property.